Relations for exact categories

Exact Categories Theo Bühler

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any em...

Exact Categories and Duality

Throughout the book Homological algebra, by H. Cartan and S. Eilenberg, the authors dealt with functors defined on categories of modules over certain rings and whose values again were modules over a ring. It will be shown in this paper that the theory may be generalized to functors defined on abstract categories, and whose values are again in such abstract categories. An abstract treatment such...

Universal Constructions for (Co)Relations: categories, monoidal categories, and props

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic...

Two Categories of Relations

Relations in Categories

This thesis investigates relations over a category C relative to an (E;M)-factorization system of C. In order to establish the 2-categoryRel(C) of relations over C in the rst part we discuss su cient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as de ...